Systematic errors in future weak-lensing surveys: requirements and prospects for self-calibration

Transcription

1 Mon. Not. R. Astron. Soc. 366, (26) doi:1.1111/j x Systematic errors in future weak-lensing surveys: requirements and prospects for self-calibration Dragan Huterer, 1 Masahiro Takada, 2 Gary Bernstein 3 and Bhuvnesh Jain 4 1 Kavli Institute for Cosmological Physics and Astronomy and Astrophysics Department, University of Chicago, Chicago, IL 6637, USA 2 Astronomical Institute, Tohoku University, Sendai , Japan 3 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 1914, USA 4 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 1914, USA Accepted 25 October 21. Received 25 October 17; in original form 25 June 2 ABSTRACT We study the impact of systematic errors on planned weak-lensing surveys and compute the requirements on their contributions so that they are not a dominant source of the cosmological parameter error budget. The generic types of error we consider are multiplicative and additive errors in measurements of shear, as well as photometric redshift errors. In general, more powerful surveys have stronger systematic requirements. For example, for a SuperNova/Acceleration Probe (SNAP)-type survey the multiplicative error in shear needs to be smaller than 1 per cent of the mean shear in any given redshift bin, while the centroids of photometric redshift bins need to be known to be better than.3. With about a factor of 2 degradation in cosmological parameter errors, future surveys can enter a self-calibration regime, where the mean systematic biases are self-consistently determined from the survey and only higher order moments of the systematics contribute. Interestingly, once the power-spectrum measurements are combined with the bispectrum, the self-calibration regime in the variation of the equation of state of dark energy is attained with only a 2 3 per cent error degradation. Keywords: cosmological parameters large-scale structure of Universe. 1 INTRODUCTION There has been significant recent progress in the measurements of weak gravitational lensing by large-scale structure. Only 5 yr after the first detections made by several groups (Bacon, Refregier & Ellis 2; Kaiser, Wilson & Luppino 2; van Waerbeke et al. 2; Wittman et al. 2), weak lensing already imposes strong constraints on the matter density relative to critical M and the amplitude of mass fluctuations σ 8 (Hoekstra, Yee & Gladders 22; Jarvis et al. 23; Heymans et al. 24; Rhodes et al. 24a; for a review see Refregier 23), as well as the first interesting constraints on the equation of state of dark energy (Jarvis et al. 25). The main advantage of weak lensing is that it directly probes the distribution of matter in the Universe. This makes weak lensing apowerful probe of cosmological parameters, including those describing dark energy (Hu & Tegmark 1999; Huterer 22; Heavens 23; Hu 23a; Refregier 23; Benabed & Van Waerbeke 24; Ishak et al. 24; Song & Knox 24; Takada & Jain 24; Takada & White 24; Ishak 25). The weak-lensing constraints are especially effective when some redshift information is available for the source galaxies; use of redshift tomography can improve the cosmological constraints by factors of a few (Hu 1999). Furthermore, dhuterer@kicp.uchicago.edu measurements of the weak-lensing bispectrum (BS) (Takada & Jain 24) and purely geometrical tests (Jain & Taylor 23; Zhang, Hui & Stebbins 23; Bernstein & Jain 24; Hu & Jain 24; Song & Knox 24; Bernstein 25) lead to significant improvements of accuracy in measuring the cosmological parameters. When these methods are combined, weak lensing by itself is expected to constrain the equation of state of dark energy w to a few per cent, and to impose interesting constraints on the time variation of w. Ongoing or planned surveys, such as the Canada France Hawaii Telescope Legacy Survey 1, the Dark Energy Survey 2 (DES), PanSTARRS 3 and Visible and Infrared Survey Telescope for Astronomy (VISTA) 4 are expected to significantly extend lensing measurements, while the ultimate precision will be achieved with the SuperNova/Acceleration Probe 5 (SNAP; Aldering et al. 24) and the Large Synoptic Survey Telescope 6 (LSST). Powerful future surveys will have very small statistical uncertainties due to the large sky coverage and huge number of galaxies, C 25 The Authors. Journal compilation C 25 RAS

2 12 D. Huterer et al. and therefore an understanding of the systematic error budget will be crucial. However, so far the rosy weak-lensing parameter accuracy predictions that have appeared in literature have not allowed for the presence of systematics [exceptions are Ishak et al. (24) and Knox et al. (25) who consider a shear calibration error, and Bernstein (25) who does the same for the cross-correlation cosmography of weak lensing]. This is not surprising, as we are just starting to understand and study the full budget of systematic errors present in weak-lensing measurements. Nevertheless, some recent work has addressed various aspects of the systematics, both experimental and theoretical, and ways to correct for them. For example, Vale et al. (24) estimated the effects of extinction on the extracted shear power spectrum (PS), while Hirata & Seljak (23), Hoekstra (24) and Jarvis & Jain (24) considered the errors in measurements of shear. Several studies explored the effects of theoretical uncertainties (Huterer et al. 24; White 24; Zhan & Knox 25; Hagan, Ma & Kravtsov 25) and ways to protect against their effects (Huterer & White 25; Huterer & Takada 25). It has been pointed out that second-order corrections in the shear predictions can be important (Cooray & Hu 22; Hamana et al. 22; Schneider, Van Waerbeke & Mellier 22; Dodelson & Zhang 25; Dodelson et al. 25; White 25). Despite these efforts, we are at an early stage in our understanding of weak-lensing systematics. Realistic assessments of systematic errors are likely to impact strategies for measuring the weaklensing shear (Bernstein 22; Bernstein & Jarvis 22; Rhodes et al. 24b; Ishak & Hirata 25; Mandelbaum et al. 25). A major effort to compare the different analysis techniques and their associated systematics is already underway (the Shear Testing Programme; Heymans et al. 25). Eventually we would like to bring weak lensing to the same level as cosmic microwave background (CMB) anisotropies and type Ia supernovae, where the systematic error budget is better understood and requirements for the control of systematic precisely outlined (e.g. Tegmark et al. 2; Hu, Hedman & Zaldarriaga 23; Kim et al. 23; Linder & Miquel 24). The purpose of this paper is to introduce the framework for the discussion of systematic errors in weak-lensing measurements and outline requirements for several generic types of systematic error. The reason that we do not consider specific sources of error (e.g. temporal variations in the telescope optics or fluctuations in atmospheric seeing) is that there are many of them, they strongly depend on a particular survey considered, and they are often poorly known before the survey has started collecting data. Instead we argue that, at this early stage of our understanding of weak-lensing systematics, it is more practical and useful to consider three generic types of error multiplicative and additive errors in measurements of shear, as well as redshift error. These generic errors are useful intermediate quantities that link actual experimental sources of error to their impact on cosmological parameter accuracy. In fact, realistic systematic errors for any particular experiment can in general be converted to these three generic systematics. Given the specifications of a particular survey, one can then estimate how much a given systematic degrades cosmological parameters. This can be used to optimize the design of the experiment to minimize the effects of systematic errors on the accuracy of desired parameter. For example, accurate photometric redshift requirements will lead to requirements on the number of filters and their wavelength coverage. Similarly, the requirements on multiplicative and additive errors in shear will determine hoccurate the sampling of the point spread function needs to be. The plan of this paper is as follows. In Section 2, we discuss the survey specifications and cosmological parameters in this study. In Section 3, we describe the parametrization of systematic errors. In Sections 4 6, we present the requirements on the systematic errors for the PS, while in Section 7 we study the requirements when both the PS and the BS are used. We combine the redshift and multiplicative errors and discuss trends in Section 8 and conclude in Section 9. 2 METHODOLOGY, COSMOLOGICAL PARAMETERS AND FIDUCIAL SURVEYS We express the measured convergence PS as Ĉ κ ij (l) = ˆP κ ij (l) + δ σγ 2 ij, (1) n i where ˆP ij κ (l) isthe measured PS with systematics [see the next section and equation (2) on how it is related to the no-systematic PS Pij κ (l)], σ 2 γ is the variance of each component of the galaxy shear and n i is the average number of resolved galaxies in the ith redshift bin per steradian. The convergence PS at a fixed multipole l and for the ith and jth redshift bins is given by P κ ij (l) = dz W i(z) W j (z) r(z) 2 H(z) P ( l r(z), z ), (2) where r(z) is the comoving angular diameter distance and H(z) is the Hubble parameter. The weights W i are given by W i (χ) = 3 2 M H 2 g i(χ)(1+ z), (3) where g i (χ) = r(χ) dχ χ s n i (χ s )r(χ s χ)/r(χ s ),χis the comoving radial distance and n i is the fraction of galaxies assigned to the ith redshift bin. We employ the redshift distribution of galaxies of the form n(z) z 2 exp( z/z ), (4) where z is survey-dependent and specified below. The cosmological constraints can then be computed from the Fisher matrix ( ) T C 1 C F ij = Cov, (5) p i p j l where C is the column vector of the observed power spectra and Cov 1 is the inverse of the covariance matrix between the power spectra whose elements are given by Cov [ Ĉ κ ij (l ), Ĉ κ (l)] kl δ ll [Ĉ κ = ik (2l + 1) f sky l (l)ĉ κ jl (l) + Ĉ κ il (l)ĉ κ (l)] jk. (6) Here l is the bandwidth in multipole we use, and f sky is the fractional sky coverage of the survey. In addition to any nuisance parameters describing the systematics, we consider six or seven cosmological parameters and assume a flat universe throughout. The six standard parameters are energy density and equation of state of dark energy DE and w, spectral index n, matter and baryon physical densities M h 2 and B h 2, and the amplitude of mass fluctuations σ 8. Note that w = constant provides useful information about the sensitivity of an arbitrary w(z), since the best-measured mode of any w(z)isabout as well measured as w = constant and therefore is subject to similar degradations in the presence of the systematics. It is this particular mode, being the most sensitive to generic systematics, that will drive the accuracy requirements we explicitly illustrate this in Fig. 2. In addition to the constant w case, we also consider a commonly used two-parameter C 25 The Authors. Journal compilation C 25 RAS, MNRAS 366,

3 Systematic errors in future weak-lensing surveys 13 Table 1. Fiducial sky coverage, density of source galaxies, variance of (each component of) shear of one galaxy, and peak of the source galaxy redshift distribution for the three surveys considered. DES SNAP LSST Area (deg 2 ) n (gal arcmin 2 ) σ γ z peak information on the physical matter and baryon densities, while the locations of the peaks help to constrain the dark energy parameters. However, we checked that, when the Planck prior added, all systematic requirements become weaker (relative to those with weak lensing alone) since the Planck information is not degraded with systematic errors even if weak-lensing information is. In order to be conservative, we decided not to add the Planck CMB information. Therefore, we consider the systematic requirements in weak-lensing surveys alone, and note that the addition of complementary information from other surveys typically weakens these requirements. description of dark energy w(z) = w + z/(1 + z) (Chevallier & Polarski 21; Linder 23) where becomes the seventh cosmological parameter in the analysis. Throughout we consider lensing tomography with 7 1 equally spaced redshift bins (see below), and we use the lensing power spectra on scales 5 l 3. We hold the total neutrino mass fixed at.1 ev; the results are somewhat dependent on the fiducial mass. We compute the linear PS using the fitting formulae of Eisenstein & Hu (1999). We generalize the formulae to w 1byreplacing the Lambda cold dark matter ( CDM) growth function of density perturbations with that for a general w(z); the latter is obtained by integrating the growth equation directly (e.g. equation 1 in Cooray, Huterer & Baumann 24). To complete the calculation of the full non-linear PS we use the fitting formulae of Smith et al. (23). The fiducial surveys, with parameters listed in Table 1, are: the Dark Energy Survey; SNAP and LSST. Note that there is some ambiguity in the definition of the number density of galaxies n g ; it is the quantity σ 2 γ /n g that determines the shear measurement noise level, where σ γ is the intrinsic shape noise of each galaxy. Galaxies implicitly assumed for the DES shear measurements are those with the largest angular sizes, and therefore they have correspondingly smaller intrinsic shape noise than the SNAP and LSST galaxies. The surveys are assumed to have the source galaxy distribution of the form in equation (4) which peaks at z peak = 2z. For the fiducial SNAP and LSST surveys, we assume tomography with 1 redshift bins equally spaced out to z = 3, as future photometric redshift accuracy will enable relatively fine slicing in redshift. For the DES, weassume a more modest seven redshift bins out to z = 2.1, reflecting the shallower reach of the DES while keeping the redshift bins equally wide ( z =.3) as in the other two surveys. Finally, we do not use weak-lensing information beyond l = 3 in order to avoid the effects of baryonic cooling (Huterer et al. 24; White 24; Zhan & Knox 24) and non- Gaussianity (White & Hu 2; Cooray & Hu 21), both of which contribute more significantly at smaller scales. While there may be ways to extend the useful l-range to smaller scales without risking bias in cosmological constraints (Huterer & White 25), extending the measurements to l max = 1 would improve the marginalized errors on cosmological parameters by only about 3 per cent. 7 The parameter fiducial values and accuracies are summarized in Table 2. The fiducial values for the parameters not listed in Table 2 are M h 2 =.147, B h 2 =.21, n = 1., and m ν =.1 ev. It is well known that measurements of the angular PS of the CMB, such as those expected by the Planck experiment, can help weak lensing to constrain the cosmological parameters. In particular, the morphology of the peaks in the CMB angular PS contains useful 7 On the other hand, especially for the DES, the Gaussian covariance assumption may be somewhat optimistic for the range 1 <l<3 (e.g. White & Hu 2). 3 PARAMETRIZATION OF THE SYSTEMATICS As mentioned above, we consider three generic sources of error. We believe that the parametrizations we propose, especially for the redshift and multiplicative shear errors, are in general enough to account for the salient effects of any generic systematic. The additive shear error is more model-dependent, and while we motivate a parametrization we believe is reasonable at this time, further theoretical and experimental work needs to be done to understand additive errors. We parametrize the redshift, multiplicative shear and additive shear errors as follows. 3.1 Redshift errors Measurements of galaxy redshifts are necessary not only for the redshift tomography which significantly improves the accuracy in measuring dark energy parameters but also to obtain the fiducial distribution of galaxies in redshift, n(z). Therefore, understanding and correcting for the redshift uncertainties is crucial, and comparison studies between currently used photometric methods, such as that initiated by Cunha et al. (in preparation), are of the highest importance. It is important to emphasize that statistical errors in photometric redshifts do not contribute to the error budget if they are well characterized. In other words, if we know precisely the distribution of photometric redshift errors (i.e. all of its moments) at each redshift, we can use the measured photometric distribution, n p (z p ), to recover the original spectroscopic distribution, n(z s )very precisely. In practice, we will not know the redshift error distribution with arbitrary precision, rather we will typically have some prior knowledge of the mean bias and scatter at each redshift. The quantity we consider in this paper is the uncalibrated redshift bias, that is, the residual (after correcting for the estimated bias) offset between the true mean redshift and the inferred mean photometric redshift at any given z. Amore general description of the redshift error would include the scatter in the redshift error at each z. Such an analysis has recently been performed by Ma, Hu & Huterer (25) who found that, even though the scatter is important as well, the mean bias in redshift is the dominant source of error. Unlike Ma et al. (25) who hold the overall distribution of galaxies in redshift n(z)fixed and only allow variations in the tomographic bin subdivisions, we allow the redshift error to affect the overall n(z) as well. At this time, it is not clear how the photometric error will affect the source galaxy distribution n(z) asitdepends on how the source galaxy distribution will be determined. We assume that the source galaxy distribution n(z)isobtained from the same photometric redshifts used to subdivide the galaxies into redshift bins. An alternative possibility is that information about the overall distribution of source galaxies is obtained from an independent source C 25 The Authors. Journal compilation C 25 RAS, MNRAS 366,

4 14 D. Huterer et al. Table 2. Cosmological parameter errors without the systematics (numbers preceding the slash in each box) and with sample systematics (numbers following the slash). For the systematics case we assumed redshift biases (described by Chebyshev polynomials) of.5, together with the multiplicative errors of.5. The errors are shown for the PS tomography only, and for the PS and the BS combined. Errors in other cosmological parameters ( M h 2, B h 2, n) are not shown. Parameter Fiducial value DES (PS) DES (PS plus BS) SNAP (PS) SNAP (PS plus BS) LSST (PS) LSST (PS plus BS) M.3.8/.11.6/.9.8/.11.4/.6.3/.7.2/.5 w 1..92/.12.35/.6.58/.81.27/.36.29/.53.1/.23 σ 8.9.1/.12.6/.8.8/.11.5/.8.4/.6.3/.4 w 1..33/.4.18/.2.28/.35.9/.12.13/.2.6/ / /.92.96/1.2.35/.44.49/.69.25/.27 (say, another survey) while the internal photometric redshifts are only used to subdivide the overall distribution into redshift bins. The two approaches, ours and that of Ma et al. (25), are therefore complementary and both should be studied. It is reassuring that the two approaches give consistent results. We consider two alternative parametrizations of the redshift error: the centroids of redshift bins, and the Chebyshev polynomial expansion of the mean bias in the z p z s relation. (i) Centroids of redshift bins. We first consider the centroid of each photometric redshift bin as a parameter. Any scatter in galaxy redshifts in a given tomographic bin will average out, to first order leaving the effect of an overall bias in the centroid of this bin. (Recall, the part of the bias that is uncalibrated and has not been subtracted out is what we consider here.) As discussed by Huterer et al. (24) in the context of number-count surveys, this approach captures the salient effects of the redshift distribution uncertainty. Note, however, that the centroid shifts do not capture the catastrophic errors where a smaller fraction of redshifts are completely mis-estimated and reside in a separate island in the z p z s plane. We therefore have B new parameters, where B is the number of redshift bins. To compute the Fisher derivatives for these parameters, we vary each centroid by some value dz, that is, we shift the whole bin by dz. As mentioned above, this procedure not only allows for the fact that the tomographic bin divisions are not perfectly measured, butitalso deforms the overall distribution of galaxies n(z). (ii) Expansion of the redshift bias in Chebyshev polynomials. While the required accuracy of redshift bin centroids provides useful information, it is sometimes difficult to compare it with directly observable quantities. In reality, an observer typically starts with measurements of photometric redshifts which are not equal to the true, spectroscopic ones: the quantity z p z s may have a non-zero value the bias and also non-zero scatter around the biased value. Therefore, it is sometimes more useful to consider requirements on the accuracy in the z p z s relation. Detailed forms of the bias and scatter are typically complicated and depend on the photometric method used and how well we are able to mimic the actual observations and correct for the biases. We write the redshift bias as a sum of N cheb smooth functions Chebyshev polynomials centred at z max /2 and extending from z = to z max, where z max (3. for SNAP and LSST, 2.1 for the DES)isthe extent of the distribution of galaxies in redshift. (We briefly review the Chebyshev polynomials in Appendix A.) The relation between the photometric and true, spectroscopic redshifts is then N cheb ( ) z p = z s + g i T i z s, (7) i=1 where z s z s z max /2, (8) z max /2 and g i are the coefficients that parametrize the bias. As with the centroids of redshift bins, we do not model the scatter in the z p z s relation as one can show that the effect of uncalibrated bias is dominant (Ma et al. 25). The effect of imperfect redshift measurements is to shift the distribution of galaxies away from the true distribution n(z s )toabiased one n p (z p ). The biased distribution of galaxies then propagates to bias the cosmological parameters. The photometric distribution can then simply be obtained from the true distribution as (e.g. Padmanabhan et al. 25) n p (z p ) = n(z) (z p z, z)dz, (9) where (z p z, z) isthe probability that the galaxy at redshift z is measured to be at redshift z p. Since we are not modelling the scatter in the z p z s relation, the probability is a delta function ( (z p z s, z s ) = δ N cheb z p z s i=1 g i T i ( z s ) ). (1) Since we will include the g i as additional parameters, with fiducial values g i =, we will need to take derivatives with one non-zero g i at a time and therefore we can assume (z p z s, z s ) = δ(z p z s g i T i (z s )) for a single i. Using the fact that δ(f(x)) = 1/ F (x ) δ(x x ) for a function F(z), where the sum runs over the roots of the equation F(x) =, and further using a recursive formula for the derivative of the Chebyshev polynomial (Arfken & Weber 2, Section 13.3), we get n p (z p ) = n(z a ) { [ ( ) 2 ]}[ ( ) ( )], a 1 + 2gi /z max 1 z a iza T i z a + iti 1 z a (11) where the sum runs over all roots of the equation z + g i T i (z ) z p =, with (recall) za (z a z max /2)/(z max /2). This is the expression for the perturbed distribution of galaxies due to a single perturbation mode. 8 We use it, together with the original distribution n(z), to compute the perturbed and unperturbed convergence power spectra and thus take the derivative with respect to g i. In Fig. 1, we show the photometric galaxy distributions n p (z p ) for the selected three Chebyshev modes (first, second and seventh) of perturbation to the distribution of galaxies n(z) corresponding to Fig. A1, together with the original unperturbed distribution. The 8 Note that, for g i 1, equation (11) simplifies to n p (z p ) = n[z p g i T i (z p )]. C 25 The Authors. Journal compilation C 25 RAS, MNRAS 366,

5 Systematic errors in future weak-lensing surveys 15 2 DES SNAP 1 1 n p (z p ) n p (z p ) z p z p Figure 1. Select three modes (first, second and seventh) of perturbation to the distribution of galaxies n(z) for the DES (left-hand panel) and SNAP (right-hand panel), shown together with the original unperturbed distribution (dashed curve in each panel). These three modes correspond to the modes of perturbation in the z p z s relation shown in Fig. A1 in Appendix A. Note that the fiducial n(z)ofsnap is broader, making the resulting wiggles in n p (z p ) less pronounced and the weak-lensing measurements therefore less susceptible to the redshift biases. Note also that the allowed perturbations to n(z) are not only located near the peak of the distribution, but can also have significant wiggles near the tails of the distribution. distributions are shown for the DES and SNAP. Since the incorrect assignment of photometric redshifts only redistributes the galaxies, we always impose the requirement n p (z p )dz p = 1regardless of the perturbation. Note that fiducial n(z) ofsnap is broader, making the resulting wiggles in n p (z p ) less pronounced and helping the weak-lensing measurements be less susceptible to the redshift biases. (As shown in Section 4.1, this effect is counteracted by the smaller fiducial errors in the SNAP survey which lead to more susceptibility to biases.) 3.2 Multiplicative errors The multiplicative error in measuring shear can be generated by a variety of sources. For example, a circular point spread function (PSF) of finite size is convolved with the true image of the galaxy to produce the observed image, and in the process it introduces a multiplicative error. Let ˆγ (z s, n) and γ (z s, n) bethe estimated and true shear of a galaxy at some true (spectroscopic) redshift z s and direction n. Then the general multiplicative factor f i (θ) acts as ˆγ (z s, n) = γ (z s, n)[1 + f i (z s, n)], (12) where f i is the multiplicative error in shear which is both directiondependent and time-dependent. We can write the error as a sum of its mean (average over all directions and redshifts in that tomographic bin) and a component with zero mean, f (z s,i, n) = f i + r i (n), (13) where f (z s,i, n) = f i and r i (n) = and the averages are taken over angle and over redshift within the ith redshift bin. Then we can write ˆγ (z s,i, n) ˆγ (z s,j, n + dn) = γ (z s,i, n) γ (z s,j, n + dn)(1+ f (z s,i, n)) (1 + f (z s, j, n)) (14) γ (z s,i, n)γ (z s, j, n + dn) (1 + f i + f j ), (15) where we dropped terms of order γγ f i f j. Since the random component of the multiplicative error is uncorrelated with shear, all terms of the form γγr are zero. Finally, terms of the form γγrr are taken to be small, thus requiring that r i (n) r j (n + dn) is smaller than f i and f j (Guzik & Bernstein 25). Therefore, ˆP κ ij (l) = Pκ ij (l)[1 + f i + f j ], (16) where again we emphasize that f i is the irreducible part of the multiplicative error in the ith redshift bin (i.e. its average over all directions and the ith redshift slab). It is clear that multiplicative errors are potentially dangerous, since they lead to an error that goes as f i and not f i 2. Finally, note that shear calibration is likely to depend on the size of the galaxy, so that the mean error f i is the error averaged over all galaxy sizes in bin i. 3.3 Additive errors Additive error in shear is generated, for example, by the anisotropy of the PSF. We define the additive error via ˆγ (z s, n) = γ (z s, n) + γ add (z s, n). (17) Let us assume that the additive error is uncorrelated with the true shears so that the term γ (n)γ add (n) is zero. Furthermore, let the Legendre transform of the term γ add (n)γ add (n) be P κ add (l). Then we can write ˆP κ ij (l) = Pκ ij (l) + Pκ add (l). (18) We now have to specify the additive systematic power P κ add (l). Consider some known sources of additive error: for example, the additive error induced due to a non-circular PSF is roughly Re PSF where e PSF is the ellipticity of the PSF and R (s PSF /s gal ) 2 is the ratio of squares of the PSF and galaxy size (E. Sheldon private communication); galaxies that are smaller therefore have a larger additive error. Of course, the observer needs to correct the overall trend in the error, leaving only a smaller residual, and the impact of this residual is what we are interested in. It should be clear from this discussion that the additive error in shear can be described by a part that depends on the average size of a galaxy at a given redshift multiplied by a random component that depends on the part of the sky observed. Therefore, we write the additive shear as γ add (z s,i, n) = b i r(n) where b i is the characteristic additive shear amplitude in the ith redshift bin and r(n)is a random fluctuation. In multipole space, the two-point function can then be expressed as b i b j times the angular part which only depends on n i n j. Note that the additive error for two galaxies at different redshifts (i.e. C 25 The Authors. Journal compilation C 25 RAS, MNRAS 366,

6 16 D. Huterer et al. when i j) isnot zero, although it may in principle be suppressed relative to the additive error for the auto-power spectra (when i = j). Motivated by such considerations, we assume the additive systematic in multipole space of the form P κ add,ij (l) = ρ b ib j ( l l ) α, (19) where the correlation coefficient ρ describes correlations between bins. The coefficient ρ is always set to unity for i = j, and for i j it is fixed to some fiducial value (not taken as a parameter in the Fisher matrix). While the discussion above would imply that ρ = 1 for all i and j,weallow for the possibility that the additive errors are not perfectly correlated across redshift bins. In the extreme case of uncorrelated additive errors in different z bins, the cross-power spectra are not affected, ρ = δ ij.wewill see that the results are weakly dependent on the fiducial value of ρ unless if ρ = 1 identically. For the multipole dependence of P κ add we assume a power-law form, and marginalize over the index α.wechoose l = 1, near the sweet spot of weak-lensing surveys, but note that this choice is arbitrary and made solely for our convenience since l is degenerate with the parameters b i. Our a priori expectation for the value of α is uncertain, and we try several possibilities but find (as discussed later) that the results are insensitive to the value of α. Wetherefore have a total of B + 1 nuisance parameters for the additive error (B parameters b i, plus α). In practice there is no particular reason why an additive systematic would be expected to conform to a power law, so in the future we should consider an additive systematic with more freedom in its spatial behaviour. The current treatment is optimistic: fixing the systematic to a power law gives it an identifying characteristic that allows us to distinguish it from cosmological signals. Without such a characteristic, the systematic could be much more damaging. 3.4 Putting the systematics together With the systematic errors we consider, the resulting theoretical convergence PS is ˆP κ ij (l) ˆP κ[ ] l, z (i) s, z ( j) s ( ) = P κ( ) α l, z (i) s + δz (i) j) l p, z( s + δz ( j) p [1 + fi + f j ] + ρ b i b j. l (2) The derivatives with respect to the multiplicative and additive parameters are trivial since they enter as linear (or power-law) coefficients, while the derivatives with respect to the redshift parameters δz (k) p (really, parameters g i defined in equation 7) need to be taken numerically. 4 RESULTS: REDSHIFT SYSTEMATICS 4.1 Centroids of redshift bins We have a total of N + B parameters, and we marginalize over the B redshift bin centroids by giving them identical Gaussian priors. While the actual redshift accuracy may be better in some redshift ranges than in others, the required accuracy that we obtain in our approach will pertain to the redshift bins where observations are most sensitive (i.e. at intermediate redshifts z.5 1). Fig. 2 shows the degradation in M, σ 8 and w = constant, and also w and, accuracies as a function of our prior knowledge of the redshift bin centroids. Here in Fig. 2 and in the subsequent two figures, the plots are shown separately for each of the three surveys (DES, SNAP and LSST). Since we are using the Fisher matrix formalism without external priors on cosmological parameters, the cosmological parameter degradations (as well as accuracies) clearly remain unchanged for an arbitrary f sky if we also scale the priors on the nuisance parameters by f 1/2 sky.inthese and subsequent plots we show the degradations for fiducial values of the sky coverage for each survey f sky,fid, where f sky,fid = f 5, f 1 and f 15 for DES, SNAP and LSST, respectively, while indicating that the requirements scale as ( f sky / f sky,fid ) 1/2. We generally find that the degradations in different cosmological parameters are comparable. To have less than 5 per cent degradation, for example, we need to control the redshift centroid bias to about.3( f sky / f sky,fid ) 1/2 for the DES and SNAP and to about.15( f sky / f sky,fid ) 1/2 for the LSST. The current statistical accuracy in individual galaxy redshifts is of the order of.2.5. Averaging the large number of galaxies in a given redshift bin, it may be that we are already close to the aforementioned requirement on the centroid bias, but this will of course depend on the control of systematics in the photometric procedure. As mentioned in the Introduction, the accurately determined combination of w and, F(w, ) w +.3,isdegraded in nearly the same way as constant w; this is illustrated by a thick dashed line in each panel of Fig. 2. In the rest of this paper we do not repeat plotting the curves corresponding to F(w, ) which nearly overlap those corresponding to w = constant. Finally, we show the degradations for w and separately. Since these two parameters are determined to a substantially lower accuracy than w = constant (see Table 2), it is not surprising that the degradations in these two parameters are substantially smaller. 4.2 Chebyshev expansion in z p z s Fig. 3 shows the degradation in the M, σ 8 and w, and also w and,aswemarginalize over a large number (N cheb = 3) of Chebyshev coefficients g i each with the given prior. We checked that, when only N 5 modes are allowed to vary, all parameters g i can be self-consistently solved from the survey this is an example of self-calibration. However, with a larger number of Chebyshev modes the self-calibration regime is lost. Since we are interested in the most general form of the redshift bias, this is the regime we would like to explore. We have also checked that, as we let the number of coefficients increase further, the errors in cosmological parameters do not increase indefinitely, as the rapid fluctuations in the z p z s are not degenerate with cosmology. In fact we found that the degradations have asymptoted to their final values once we include 2 3 Chebyshev polynomials; therefore, fluctuations in redshift on scales smaller than z.1 are not degenerate with cosmological parameters. The choice of N cheb = 3 coefficients is therefore conservative. Fig. 3 shows that the photometric redshift bias in each Chebyshev mode, for the DES and SNAP, should be controlled to about.1( f sky / f sky,fid ) 1/2 or less. For LSST, the requirement is more severe and we could tolerate biases no larger than about.5 ( f sky / f sky,fid ) 1/2. These are fairly stringent requirements, in rough agreement with results found by Ma et al. (25). It might seem a bit surprising that the SNAP and DES requirements are comparable, given that the fiducial SNAP survey is more powerful than the DES (see Table 2) and hence would need better control of the systematics. However, there is another effect at play here: a narrow distribution of galaxies in redshift, such as that from C 25 The Authors. Journal compilation C 25 RAS, MNRAS 366,

7 Systematic errors in future weak-lensing surveys DES Ω σ M 8 w or F(w, ) w SNAP Ω M σ 8 w or F(w, ) w prior on centroids of redshift bins prior on centroids of redshift bins LSST Ω M σ 8 w or F(w, ) w prior on centroids of redshift bins Figure 2. Degradation in the cosmological parameter accuracies as a function of our prior knowledge of δz z p z s.weassume equal Gaussian priors to each redshift bin centroid, shown on the x-axis. For example, to have less than 5 per cent degradation in M, σ 8 or w,weneed to control the redshift bias to about.3 ( f sky / f sky,fid ) 1/2 or better for the DES and SNAP, and to about.15 ( f sky / f sky,fid ) 1/2 or better for the LSST. For the varying equation-of-state parametrization, the requirements for the best-measured combination of w and (F(w, ) w +.3 ) are identical to those for w = constant, while the requirements on w and individually are somewhat less stringent. the DES,ismore strongly subjected to fluctuations in z p z s than a wide distribution, such as that from SNAP; (see Fig. 1). Therefore, surveys with wide leverage in redshift have an advantage in beating down the effect of redshift error, just as in the case of cluster count surveys (Huterer et al. 24). We now confirm the results by computing the bias in the cosmological parameters due to the bias in one of the redshift bias coefficients, dg i. The bias in the cosmological parameter p α can be computed as δ p α = F 1 αβ l [Ĉ κ i (l) C κ i (l)] Cov 1 [ C κ i (l), C κ j (l)] C j (l) κ p β, (21) where the summations over β, i and j were implicitly assumed and the covariance of the cross-power spectra is given in equation (6). The source of the bias in the observed shear covariance, Ĉi κ(l) C i κ(l), is assumed to be the excursion of dg m =.1 in the mth coefficient of the Chebyshev expansion, others being held to their fiducial values of zero. We then compare this error to the 1σ marginalized error on the parameter p α.for a range of 1 m 3 we find that the biases in the cosmological parameters are entirely consistent with statistical degradations shown in Fig. 3. We also find that the degradations due to higher modes (m 1 2) are progressively suppressed, illustrating again that smooth biases in redshift are the most important sources of degeneracy with cosmological parameters, and need the most attention when calibrating the photometric redshifts. Finally, let us note that in this analysis we have not attempted to model more complicated redshift dependences of bias, the presence of abrupt degradations in redshift etc. Such an analysis is beyond the scope of this work, but can be done using similar tools once the details about the performance of a given photometric method are known. 5 RESULTS: MULTIPLICATIVE ERROR As explained in Section 3.2, we adopt the multiplicative error of the form ˆP κ ij (l) = Pκ ij (l)[1 + f i + f j ], (22) where ˆP ij κ(l) and Pκ ij (l) are estimated and true convergence power, respectively. In other words, the multiplicative error we consider can be thought of as an irreducible but perfectly coherent bias in the calibration of shear in any given redshift bin. C 25 The Authors. Journal compilation C 25 RAS, MNRAS 366,

8 18 D. Huterer et al DES w Ω M w σ SNAP Ω M w σ 8 w prior on Chebyshev coefficients g i prior on Chebyshev coefficients g i σ 8 w LSST Ω M w prior on Chebyshev coefficients g i Figure 3. Degradation in marginalized errors in M, σ 8 and w = constant, as well as w and,asafunction of our prior knowledge of the redshift bias coefficients g i for the DES, SNAP and LSST. We use N cheb = 3 parameters g i that describe the bias in redshift and give equal prior to each of them, shown on the x-axis. For the DES and SNAP, knowledge of g i to better than.1( f sky / f sky,fid ) 1/2, corresponding to redshift bias of z p z s.1( f sky / f sky,fid ) 1/2 for each Chebyshev mode, is desired as it leads to error degradations of about 5 per cent or less. For LSST the requirement is about a factor of 2 stronger. These results are corroborated by computing the bias in cosmological parameters as discussed in the text. First, note that a tomographic measurement with B redshift bins can determine at most B multiplicative parameters f i (this is true either if they are just step-wise excursions as assumed here or coefficients of Chebyshev polynomials described in the Appendix A). For, if we had more than B multiplicative parameters, at least one bin would have two or more parameters, and there would be an infinite degeneracy between them. In contrast, the survey can in principle determine a much larger number of cosmological parameters since they enter the observables in a more complicated way. We choose to use exactly N mult = B multiplicative parameters (so N mult = 1 for SNAP and LSST and N mult = 7 for the DES) since we would expect the shear calibration to be different within each redshift bin. Fig. 4 shows the degradation in error in measuring three cosmological parameters as a function of our prior knowledge of the multiplicative factors; we give equal prior to all multiplicative factors. For the DES and SNAP, control of multiplicative error of.1(f sky /f sky,fid ) 1/2 (or 1 per cent in shear for the fiducial sky coverages) leads to a 5 per cent increase in the cosmological error bars, and is therefore about the largest error tolerable. For LSST, the requirement is about a factor of 2 more stringent. In general, it is interesting to consider whether the weak-lensing survey can self-calibrate, that is, determine both the cosmological and the nuisance parameters concurrently. This is partly motivated by the self-calibration of cluster count surveys, where it has been shown that one can determine the cosmological parameters and the evolution of the mass-observable relation provided that the latter takes a relatively simple deterministic form (e.g. Levine, Schulz & White 22; Hu 23b; Majumdar & Mohr 23; Lima & Hu 24). Fig. 4 shows that the fiducial DES and SNAP surveys can self-calibrate with about a 1 per cent degradation in cosmological parameter errors (and LSST with about a 15 per cent degradation). While doubling the error in cosmological parameters is a somewhat steep price to pay, it is very encouraging that all surveys enter a self-calibrating regime where only the higher-order moments of the error contribute to the total error budget. In Section 7, we show that the inclusion of BS information can significantly improve the self-calibration regime. 6 RESULTS: ADDITIVE ERROR As explained in Section 3.3, we adopt the additive error of the form P κ add,ij (l) = ρb ib j ( l l ) α, (23) C 25 The Authors. Journal compilation C 25 RAS, MNRAS 366,

9 Systematic errors in future weak-lensing surveys DES 14 SNAP Ω M w σ8 w w a w σ 8 Ω M w prior on multiplicative factors in shear prior on multiplicative factors in shear LSST Ω M σ 8 w w prior on multiplicative factors in shear Figure 4. Degradation in marginalized errors in M, σ 8 and w = constant, as well as w and,asafunction of our prior knowledge of the shear multiplicative factors. We give equal prior to multiplicative factors in all redshift bins, and show results for the DES, SNAP and LSST. For example, existence of the multiplicative error of.1( f sky / f sky,fid ) 1/2 (or 1 per cent in shear for the fiducial sky coverages) in each redshift bin leads to 5 per cent increase in error bars on M, σ 8 and w for the DES and SNAP, and about a 1 per cent degradation for LSST. which adds that amount of noise to the convergence power spectra. The coefficient ρ is always 1 for i = j, and its (fixed) value for i j controls how much additive error leaks into the cross-power spectra. We weigh the fiducial value of b i by the inverse square of the average galaxy size in the ith redshift bin (or, by the square of the angular diameter distance to the ith bin). 9 Fig. 5 shows the degradations in the equation-of-state s a function of the fiducial b i at redshift z =.75 where, very roughly, most galaxies are found (recall, the other b i are equal to this value modulo order-unity weighting by the square of the angular diameter distance to their corresponding redshift). The solid line in the figure shows results for our fiducial SNAP survey, assuming no contribution to the cross-power spectra (i.e. ρ = δ ij ). The coefficients b i need to be controlled to 2 1 5, corresponding to shear variance of l(l + 1)P κ add,ij (l)/(2π) 1 4 on scales of 1 arcmin (l 1) where most constraining power of weak lensing resides. The observed degradation in cosmological parameters is clearly due to the increased sample variance that the additional power puts on to the measurement of the PS. When ρ = 1, this sample variance is 9 For b i =, the Fisher derivatives with respect to b i are zero and no information about these parameters can formally be extracted. confined to a single mode of the shear covariance matrix, so that the maximum damage is limited and we observe the self-calibration plateau in Fig We have tried varying a number of other details. (i) Using different values of the coefficient ρ for i j in the range ρ<1; (ii) changing the fiducial value of the exponent α from to +3 or 3; (iii) adding a 1 per cent prior to the b i (rather than no prior); (iv) using the redshift-independent fiducial values of the coefficients b i and (v) considering the degradation in the other cosmological parameters. Interestingly, we find that the overall requirements are very weakly dependent on any of the above variations, and the 1 One could in principle also have a worst-case limit with additive errors whose functional form makes them strongly degenerate with the effect of varying the cosmological parameters, where the accuracy in cosmological constraints degrades as soon as the additive power becomes comparable to the measurement uncertainties in the PS (and not the PS itself, as above). C 25 The Authors. Journal compilation C 25 RAS, MNRAS 366,

10 11 D. Huterer et al. % degradation in σ w Cross-power correlation: % 9% 99% 1% fiducial value of b i at z=.75 Figure 5. Degradation in the cosmological parameter accuracies as a function of the fiducial value of the additive shear errors b i, assuming no prior on the b i.weshow results for SNAP and for several values of the correlation coefficients between different bins, ρ, and marginalizing over the spatial power-law exponent α which has fiducial value α =. The results are insensitive to various details, as discussed in the text, and the only exception is the possibility that the additive effect on the cross-power spectra is negligibly suppressed (i.e. the cross-power correlation is close to 1 per cent). We conclude that the mean additive shear will need to be known to about 2 1 5, corresponding to shear variance of 1 4 on scales of 1 arcmin. requirements almost always look roughly like those in Fig. 5. In particular, the results are essentially insensitive to reasonable priors of any of the nuisance parameters, since the b i can be determined internally to an accuracy much better than b i for all but the smallest ( 1 7 ) fiducial values of these parameters. Therefore, the dominant effect by far is the fiducial value of b i and the increased sample variance that it introduces. The only interesting exception to this insensitivity is allowing the i j value of the coefficient ρ to be very close to unity. In the extreme case when ρ = 1 for i j, the degradation asymptotes to 15 per cent even with very large b i since in that case the additive errors add a huge contribution to all power spectra, and one can mathematically show that the resulting errors in cosmological parameters do not change more than 1 per cent irrespective of the systematic error. In practice, however, ρ.99 for i j is needed to see an appreciable difference (see the other curves in Fig. 5), and it is reasonable to believe that the actual errors in the cross-power spectra will be suppressed by much more than a per cent, resulting in the degradations as shown by the solid curve in Fig. 5. While our model for the additive errors is admittedly crude, it is very difficult to parametrize these errors more accurately without end-to-end simulations that describe various systematic effects and estimate their contribution to the additive errors (such simulations are now being planned or carried out by several research groups). Moreover, additive errors cannot be self-calibrated unless we can identify a functional dependence, which is distinct from the cosmological dependence of the PS, that it must have. Finally, we note that space-based surveys, such as SNAP, are expected to have a more accurate characterization of the additive error, primarily from the absence of atmospheric effects in the characterization of the PSF. 7 SYSTEMATICS WITH THE BISPECTRUM We now extend the calculations to the BS of weak gravitational lensing. The BS is the Fourier counterpart of the three-point correlation function, and it describes the non-gaussianity of mass distribution in the large-scale structure that is induced by gravitational instability from the primordial Gaussian perturbations that the simplest inflationary models predict. The redshift evolution and configuration dependence of the mass BS can be accurately predicted using a suite of high-resolution N-body simulations (see e.g. Bernardeau et al. 22, for a comprehensive review). The BS of lensing shear arises from the line-of-sight integration of products of the mass BS and the lensing geometrical factor (see equation 18 in Takada & Jain 24). Following Jain & Seljak (1997), we roughly estimate how the lensing PS and BS scale with cosmological parameters by perturbing around the fiducial CDM model: P κ 3.5 DE σ z 1.6 s w.31, B κ 6.1 DE σ z 1.6 s w.19, (24) where we have considered multipole mode of l = 1, equilateral triangle configurations in multipole space, redshift of all source galaxies z s = 1 (no tomography), and constant equation-of-state parameter w. Weadopt the model described in Takada & Jain (24) to compute the lensing BS. Equation (24) illustrates that the PS and the BS depend upon specific combinations of cosmological parameters. For example, the PS (and more generally any two-point statistics of choice) depends on DE and σ 8 through the combination 1.2 DE σ 8 (or equivalently.5 M σ 8). Furthermore, the cosmological parameter dependences are strongly degenerate with source redshift (z s ) unless accurate redshift information is available. Importantly, the PS and BS have substantially different dependences on the parameters, suggesting that combining the two can be a powerful way of breaking the parameter degeneracies. For example, it is well known that a combination of B/P 2, motivated from the hierarchical clustering ansatz, depends mainly on DE, with rather weak dependence on σ 8,sothat roughly B/P 2.9 DE σ.1 8 (Bernardeau, Van Waerbeke & Mellier 1997; Hui 1999; Takada & Jain 24). As another example, the BS amplitude increases with the mean source redshift z s more slowly than P 2 because the non-gaussianity of structure formation becomes suppressed at higher redshifts. It is therefore clear that photometric redshift errors of source galaxies will affect the PS and BS differently. Likewise, it is natural to expect that other systematics that we have considered affect the PS and the BS in a different way. For the PS plus BS systematic analysis, we consider the redshift and multiplicative errors but not the additives. For the redshift errors we adopt the parametrization of redshift bin centroids, nopplied to tomographic bins of both PS and BS. The multiplicative error is modelled similarly as in Section 3.2: the observed BS with tomographic redshift bins i, j and k, ˆB ijk (l 1, l 2, l 3 ), is related to the true BS B ijk (l 1, l 2, l 3 ) via ˆB ijk (l 1, l 2, l 3 ) = B ijk (l 1, l 2, l 3 )[1 + f i + f j + f k ]. (25) Based on the considerations above, we study how combining the PS with the BS allows us to break parameter degeneracies not only in cosmological parameter space but also those induced by the presence of the nuisance systematic parameters. We include all triangle configurations, and compute the bispectra constructed using five redshift bins up to l max = 3. Note that, for a given triangle configuration, there are 5 3 tomographic bispectra; the large amount of time necessary to compute them is the reason why we C 25 The Authors. Journal compilation C 25 RAS, MNRAS 366,